Lemma 67.34.5. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S Let n \geq 0. Assume f is locally of finite presentation. The open
W_ n = \{ x \in |X| \text{ such that the relative dimension of }f\text{ at } x \leq n\}
of Lemma 67.34.4 is retrocompact in |X|. (See Topology, Definition 5.12.1.)
Proof.
Choose a diagram
\xymatrix{ U \ar[r]_ h \ar[d]_ a & V \ar[d] \\ X \ar[r] & Y }
where U and V are schemes and the vertical arrows are surjective and étale, see Spaces, Lemma 65.11.6. In the proof of Lemma 67.34.4 we have seen that a^{-1}(W_ n) = U_ n is the corresponding set for the morphism h. By Morphisms, Lemma 29.28.6 we see that U_ n is retrocompact in U. The lemma follows by definition of the topology on |X|, compare with Properties of Spaces, Lemma 66.5.5 and its proof.
\square
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